If I understand, you're asking two questions. To the first, launching by consulting. If you have no experience, you're basically a teacher telling the client everything you learned in school because they could go to school like you did but they'd be farther ahead having context. Since they likely don't have the time to go, you're just passing along what you selectively edited as applicable to them, culling from your coursework. Nowhere is this clearer than in manufacturing. Newly minted MBAs never get into so much trouble as when they tackle that.

As to the second, should you gain broad experience or focus solely on your intended industry. That is a tough one. If you're not in love with your intended industry, shop around. You may find something in the oddest place that you'll love as much as it will love you. If you love your intended industry, go for it. Give it your all, there's always room for somebody who is good, passion trumps. I'm probably a one-trick pony from the perspective of most but the farther you get into it, the more specialization you discover there is.

I'm pretty specialized. I fell in love with my career right away, thought it would be the finest thing in the world. I still love what I do after 3 decades and haven't done much else beyond occasional forays. It doesn't keep me from learning about other things that find their ways into what I do.